Mathematics

In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long 'search' for a purely mathematical incompleteness result in first-order arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse 2^\omega to \omega, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size ℵ α , then the set has size ℵ α if ℵ α is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If G=( V G , E G ) is a connected locally finite chordal graph, then there is an ordering < of V G such that {w<v:{w,v}∈ E G } is a clique for each v∈ V G .

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find L ω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences. Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. Our advance here stems from a careful analysis of the Halpern-Laüchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken's tree theorem that permits us to gauge its effectivity in turn. The principal outcome of this is a comprehensive classification of the computable content of Milliken's tree theorem. We apply our analysis also to several well-known applications of Milliken's tree theorem, namely Devlin's theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey's theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken's tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. We identify again the familiar dichotomy between coding the halting problem or not based on the size of instance, but this is more subtle here owing to the more complicated underlying structures, particularly in the case of Devlin's theorem. We also establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker's notion of big Ramsey structure.

In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke's possible world semantics. This kind of modal systems characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70's. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several well-known controversial issues of quantified modal logics, relative to the identity predicate, Barcan's formulas, and de dicto and de re modalities, can be tackled from a new angle within the present framework.

A topological space is \emph{hereditarily k -irresolvable} if none of its subspaces can be partitioned into k dense subsets, We use this notion to provide a topological semantics for a sequence of modal logics whose n -th member K4 C n is characterised by validity in transitive Kripke frames of circumference at most n . We show that under the interpretation of the modality ◊ as the derived set (of limit points) operation, K4 C n is characterised by validity in all spaces that are hereditarily n+1 -irresolvable and have the T D separation property. We also identify the extensions of K4 C n that result when the class of spaces involved is restricted to those that are weakly scattered, or crowded, or openly irresolvable, the latter meaning that every non-empty open subspace is 2-irresolvable. Finally we give a topological semantics for K4M, where M is the McKinsey axiom.

This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic {\textbf{IL}} . Next, the method is applied to obtain new results: the modal completeness of the logic {\textbf{IL}}{\sf M_0} , and modal completeness of {\textbf{IL}}({\sf W^*}) .

We study modal completeness and incompleteness of several sublogics of the interpretability logic IL . We introduce the sublogic IL − , and prove that IL − is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between IL − and IL with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of IL are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to generalized Veltman prestructures. As a consequence of these investigations, we obtain that the twenty logics studied in this paper are all decidable.

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written ◊φ ) if φ is true in some extension of that structure, and φ is necessary (written □φ ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) of all models of a given theory T---all graphs, all groups, all fields, or what have you---considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k -colorability, finiteness, countability, size continuum, size ℵ 1 , ℵ 2 , ℵ ω , ℶ ω , first ℶ -fixed point, first ℶ -hyper-fixed-point and much more. A graph obeys the maximality principle ◊□φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.